102 Chapter 2. Solution of Linear Algebraic Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copying of machine- readable files (including this one) to any servercomputer, is strictly prohibited. To order Numerical Recipes books,diskettes, or CDROMs visit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America). We will make use of QR decomposition, and its updating, in §9.7. CITED REFERENCES AND FURTHER READING: Wilkinson, J.H., and Reinsch, C. 1971, Linear Algebra ,vol.IIof Handbook for Automatic Com- putation (New York: Springer-Verlag), Chapter I/8. [1] Golub, G.H., and Van Loan, C.F. 1989, Matrix Computations , 2nd ed. (Baltimore: Johns Hopkins University Press), §§ 5.2, 5.3, 12.6. [2] 2.11 Is Matrix Inversion an N 3 Process? We close this chapter with a little entertainment, a bit of algorithmic prestidig- itation which probes more deeply into the subject of matrix inversion. We start with a seemingly simple question: How many individual multiplications does it take to perform the matrix multiplication of two 2 × 2 matrices,  a 11 a 12 a 21 a 22  ·  b 11 b 12 b 21 b 22  =  c 11 c 12 c 21 c 22  (2.11.1) Eight, right? Here they are written explicitly: c 11 = a 11 × b 11 + a 12 × b 21 c 12 = a 11 × b 12 + a 12 × b 22 c 21 = a 21 × b 11 + a 22 × b 21 c 22 = a 21 × b 12 + a 22 × b 22 (2.11.2) Do you think that one can write formulas for the c’s that involve only seven multiplications? (Try it yourself, before reading on.) Such a set of formulas was, in fact, discovered by Strassen [1] . The formulas are: Q 1 ≡ (a 11 + a 22 ) × (b 11 + b 22 ) Q 2 ≡ (a 21 + a 22 ) × b 11 Q 3 ≡ a 11 × (b 12 − b 22 ) Q 4 ≡ a 22 × (−b 11 + b 21 ) Q 5 ≡ (a 11 + a 12 ) × b 22 Q 6 ≡ (−a 11 + a 21 ) × (b 11 + b 12 ) Q 7 ≡ (a 12 − a 22 ) × (b 21 + b 22 ) (2.11.3) 2.11 Is Matrix Inversion an N 3 Process? 103 Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copying of machine- readable files (including this one) to any servercomputer, is strictly prohibited. To order Numerical Recipes books,diskettes, or CDROMs visit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America). in terms of which c 11 = Q 1 + Q 4 − Q 5 + Q 7 c 21 = Q 2 + Q 4 c 12 = Q 3 + Q 5 c 22 = Q 1 + Q 3 − Q 2 + Q 6 (2.11.4) What’s the use of this? There is one fewer multiplication than in equation (2.11.2), but many more additions and subtractions. It is not clear that anything has been gained. But notice that in (2.11.3) the a’s and b’s are never commuted. Therefore (2.11.3) and (2.11.4) are valid when the a’s and b’s are themselves matrices. The problem of multiplyingtwo very large matrices (of order N =2 m for some integer m) can now be broken down recursively by partitioning the matrices into quarters, sixteenths, etc. And note the key point: The savings is not just a factor “7/8”; it is that factor at each hierarchical level of the recursion. In total it reduces the process of matrix multiplication to order N log 2 7 instead of N 3 . What about all the extra additions in (2.11.3)–(2.11.4)? Don’t they outweigh the advantage of the fewer multiplications? For large N, it turns out that there are six times as many additions as multiplications implied by (2.11.3)–(2.11.4). But, if N is very large, this constant factor is no match for the change in the exponent from N 3 to N log 2 7 . With this “fast” matrix multiplication,Strassen also obtained a surprising result for matrix inversion [1] . Suppose that the matrices  a 11 a 12 a 21 a 22  and  c 11 c 12 c 21 c 22  (2.11.5) are inverses of each other. Then the c’s can be obtained from the a’s by the following operations (compare equations 2.7.22 and 2.7.25): R 1 = Inverse(a 11 ) R 2 = a 21 × R 1 R 3 = R 1 × a 12 R 4 = a 21 × R 3 R 5 = R 4 − a 22 R 6 = Inverse(R 5 ) c 12 = R 3 × R 6 c 21 = R 6 × R 2 R 7 = R 3 × c 21 c 11 = R 1 − R 7 c 22 = −R 6 (2.11.6) 104 Chapter 2. Solution of Linear Algebraic Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copying of machine- readable files (including this one) to any servercomputer, is strictly prohibited. To order Numerical Recipes books,diskettes, or CDROMs visit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America). In (2.11.6) the “inverse” operator occurs just twice. It is to be interpreted as the reciprocal if the a’s and c’s are scalars, but as matrix inversion if the a’s and c’s are themselves submatrices. Imagine doing the inversionof a very large matrix, of order N =2 m , recursively by partitions in half. At each step, halving the order doubles the number of inverse operations. But this means that there are only N divisions in all! So divisions don’t dominate in the recursive use of (2.11.6). Equation (2.11.6) is dominated, in fact, by its 6 multiplications. Since these can be done by an N log 2 7 algorithm, so can the matrix inversion! This is fun, but let’s look at practicalities: If you estimate how large N has to be before the difference between exponent 3 and exponent log 2 7=2.807 is substantial enough to outweigh the bookkeeping overhead, arising from the complicated nature of the recursive Strassen algorithm, you will find that LU decomposition is in no immediate danger of becoming obsolete. If, on the other hand, you like this kind of fun, then try these: (1) Can you multiplythecomplex numbers (a+ib) and (c +id) in only threereal multiplications? [Answer: see §5.4.] (2) Can you evaluate a general fourth-degree polynomial in x for many different values of x with only three multiplications per evaluation? [Answer: see §5.3.] CITED REFERENCES AND FURTHER READING: Strassen, V. 1969, Numerische Mathematik , vol. 13, pp. 354–356. [1] Kronsj¨o, L. 1987, Algorithms: Their Complexity and Efficiency , 2nd ed. (New York: Wiley). Winograd, S. 1971, Linear Algebra and Its Applications , vol. 4, pp. 381–388. Pan, V. Ya. 1980, SIAM Journal on Computing , vol. 9, pp. 321–342. Pan, V. 1984, How to Multiply Matrices Faster , Lecture Notes in Computer Science, vol. 179 (New York: Springer-Verlag) Pan, V. 1984, SIAM Review , vol. 26, pp. 393–415. [More recent results that show that an exponent of 2.496 can be achieved — theoretically!] . 102 Chapter 2. Solution of Linear Algebraic Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5). = −R 6 (2.11.6) 104 Chapter 2. Solution of Linear Algebraic Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)